We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space where . This result was obtained in [Kenig-Ponce-Vega, 2000] following Bourgain's method ([Bourgain, 1998]). We present here a different and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations ([Calderon, 1990], [Gallagher-Planchon, 2002]).
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Isabelle Gallagher, Fabrice Planchon, On global solutions to a defocusing semi-linear wave equation. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 161–177DOI 10.4171/RMI/341